Upper limit to the maximum cumulative reward in a deep reinforcement learning problem
In any reinforcement learning problem, not just Deep RL, then there is an upper bound for the cumulative reward, provided that the problem is episodic and not continuing.
If the problem is episodic and the rewards are designed such that the problem has a natural ending, i.e. the episode will end regardless of how well the agent does in the environment, then then you could work it out by calculating the max possible reward in each step of the episode; however this is potentially non-trivial depending on your environment.
For an example in a trivial setting, however, imagine the problem of cartpole -- I could define the MDP to have a reward of +1 for every time step that the agent is able to balance the pole upright, and 0 when the pole falls. If I also defined that the problem terminates after 200 time steps then the upper bound on cumulative rewards for this problem would be 200.
In general, if the problem is continuing then in theory the problem goes on infinitely and so there is no upper bound, as the episode never ends -- this is partly why we use the discount factor, to ensure that $\sum_{k=0} \gamma^k R_{t+k}$ converges.